I’d like to begin the use of cloudflare on “Free” plan and the system gives me specific nameservers that I need to set for my domain.
I don’t like the name of these nameservers. Is it possible that in a day or two if once again I’d like to add my domain into the cloudflare account - the system will provide me with different nameservers.
You do have to use the provided nameservers. You can add the domain to a different account and may get different nameservers, but that is not guaranteed. Otherwise, you could only use custom nameservers or a CNAME setup. Both require a Business plan.